Left Quasi- ArtinianModules

Pacific Journal of Mathematics

Hacettepe Journal of Mathematics and Statistics, 2017

In this work we give sufficient conditions for a ring R to be quasi-Frobenius, such as R being left artinian and the class of injective cogenerators of R-Mod being closed under projective covers. We prove that R is a division ring if and only if R is a domain and the class of left free R-modules is closed under injective hulls. We obtain some characterizations of artinian principal ideal rings. We characterize the rings for which left cyclic modules coincide with left cocyclic R-modules. Finally, we obtain characterizations of left artinian and left coartinian rings.

Journal of Algebra, 1985

The aim of this paper is to investigate generalizations of locally artinian supplemented modules in module theory, namely locally artinian radical supplemented modules and strongly locally artinian radical supplemented modules. We have obtained elementary features of them. Also, we have characterized strongly locally artinian radical supplemented modules by left perfect rings. Finally, we have proved that the reduced part of a strongly locally artinian radical supplemented R-module has the same property over a Dedekind domain R.

Proceedings of The Edinburgh Mathematical Society, 1989

It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right i?-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right /^-module is injective. This leads to the second general question: given a ring R all of whose cyclic right /^-modules have a certain property, what can one say about R itself?

Journal of Mathematical Sciences, 2009

All right R-modules are I0-modules if and only if either R is a right SV-ring or R/I (2) (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I (2) (R) is equal to zero. All rings are assumed to be associative and with nonzero identity element. Expressions such as "an Artinian ring" mean that the corresponding right and left conditions hold. A submodule X of the module M is said to be superfluous in M if X + Y = M for every proper submodule P of the module M. Following [9], we call a module M an I 0-module if every nonsuperfluous submodule of M contains a nonzero direct summand of the module M. It is clear that I 0-modules are weakly regular modules, considered in [1-3, 8]; a module M is said to be weakly regular if every submodule of M that is not contained in the Jacobson radical of M contains a nonzero direct summand of M. Weakly regular modules are studied in [1-3; 6; 8; 9; 11, Chap. 3; 12-14], and other papers. A ring R is called a right generalized SV-ring if every right R-module is weakly regular. It is clear that A is a right generalized SV-ring provided each right A-module is an I 0-module. In addition, it follows from the presented paper that the Jacobson radical of any right module over a right generalized SV-ring is superfluous; therefore, every right module over a right generalized SV-ring is an I 0-module. The aim of the paper is the study of generalized right SV-rings. The main result of the present paper is Theorem 1.

Journal of Mathematics Research, 2012

Let R be a non-necessarily commutative ring and M an R-module. We use the category σ[M] to introduce the notion of I-module who is a generalization of I-ring. It is well known that every artinian object of σ[M] is cohopfian but the converse is not true in general. The aim of this paper is to characterize for a fixed ring, the left (right) R-modules M for which every co-hopfian object of σ[M] is artinian. We obtain some characterization of finitely generated I-modules over a commutative ring, faithfully balanced finitely generated I-modules, and left serial finitely generated I-modules over a duo-ring.

Journal of Algebra, 1985

Lavoro eseguito nell'ambito dell'attivit8 dei gruppi di ricerca matematica de1 CNR. 415 476 MENINI AND ORSATTI Let g* be the class of r-torsion modules: KZ = {ME R-Mod: V x E M, Ann,(x) E ;T},

Journal of Pure and Applied Algebra, 2004

A module M is said to satisfy the condition (˝ *) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ *) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right˝ *-semisimple rings. Right˝ *-semisimple rings are right artinian. However, in general, a right˝ *-semisimple rings need not be left˝ *-semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left˝ *-semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.

Publicacions Matemàtiques, 2013

We carry out a study of rings R for which Hom R (M, N) = 0 for all nonzero N ≤ M R. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.